On the double density spectra of compact spaces

Abstract

The set $dd(X)$ of densities of all dense subspaces of a topological space $X$ is called the double density spectrum of $X$. In this note we present a couple of results that imply $\lambda \in dd(X)$, provided that $X$ is a compact space and $\lambda$ is a cardinal satisfying certain conditions. As a consequence of these results, we prove that $dd(X) = [d(X), w(X)]$ holds for any polyadic space $X$. This, in turn, implies that $dd(G) = [d(G), w(G)]$ for any locally compact topological group $G$.